From 94f7eb74969c6d3c98775dd1c8765df771f0c33f Mon Sep 17 00:00:00 2001 From: endolith Date: Mon, 11 Jul 2022 22:14:12 -0400 Subject: [PATCH 1/4] Add variance error bars to CE example https://stats.stackexchange.com/questions/576491/how-to-calculate-the-accuracy-of-a-yes-no-monte-carlo-simulation-such-as-throwi#comment1064757_576491 --- examples/merrill_1984_fig_2c_2d.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/examples/merrill_1984_fig_2c_2d.py b/examples/merrill_1984_fig_2c_2d.py index 114586c..51e296b 100644 --- a/examples/merrill_1984_fig_2c_2d.py +++ b/examples/merrill_1984_fig_2c_2d.py @@ -140,7 +140,8 @@ 'Plurality'): x, y = zip(*sorted(condorcet_winner_count[method].items())) CE = np.array(y)/y_cw - plt.plot(x, CE*100, '-', label=method) + var = CE*(1-CE)/y_cw # If p = N/n, then var = p(1-p)/n + plt.errorbar(x, CE*100, var*100, fmt='-', label=method) table.append([method, *CE*100]) print(tabulate(table, ["Method", *x], tablefmt="pipe", floatfmt='.1f')) From 24aee1382786c9a2e1080f689db4fa6e3420dc60 Mon Sep 17 00:00:00 2001 From: endolith Date: Wed, 20 Jul 2022 00:18:02 -0400 Subject: [PATCH 2/4] Use 95% confidence interval for error bars instead https://stackoverflow.com/a/73045745/125507 --- examples/merrill_1984_fig_2c_2d.py | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/examples/merrill_1984_fig_2c_2d.py b/examples/merrill_1984_fig_2c_2d.py index 51e296b..51021c3 100644 --- a/examples/merrill_1984_fig_2c_2d.py +++ b/examples/merrill_1984_fig_2c_2d.py @@ -44,6 +44,7 @@ import time from collections import Counter import numpy as np +from scipy.stats import binomtest import matplotlib.pyplot as plt from tabulate import tabulate from elsim.methods import (fptp, runoff, irv, approval, borda, coombs, @@ -140,8 +141,10 @@ 'Plurality'): x, y = zip(*sorted(condorcet_winner_count[method].items())) CE = np.array(y)/y_cw - var = CE*(1-CE)/y_cw # If p = N/n, then var = p(1-p)/n - plt.errorbar(x, CE*100, var*100, fmt='-', label=method) + ci = np.empty((2, len(y))) + for i in range(len(y)): + ci[:, i] = binomtest(y[i], y_cw[i]).proportion_ci() + plt.errorbar(x, CE*100, ci*100, fmt='-', label=method) table.append([method, *CE*100]) print(tabulate(table, ["Method", *x], tablefmt="pipe", floatfmt='.1f')) From 5bd979cad70e9c001275ee683a8d14ef16a746bf Mon Sep 17 00:00:00 2001 From: endolith Date: Wed, 20 Jul 2022 00:23:03 -0400 Subject: [PATCH 3/4] Convert error from absolute to relative proportion_ci outputs absolute values, but plt.errorbar requires values relative to the original data. Update results images --- examples/merrill_1984_fig_2c_2d.py | 6 +++++- ...gure_2.c._201_voters,_10000_iterations.png | Bin 77338 -> 77112 bytes ...gure_2.d._201_voters,_10000_iterations.png | Bin 76911 -> 76791 bytes 3 files changed, 5 insertions(+), 1 deletion(-) diff --git a/examples/merrill_1984_fig_2c_2d.py b/examples/merrill_1984_fig_2c_2d.py index 51021c3..98bf8cc 100644 --- a/examples/merrill_1984_fig_2c_2d.py +++ b/examples/merrill_1984_fig_2c_2d.py @@ -141,10 +141,14 @@ 'Plurality'): x, y = zip(*sorted(condorcet_winner_count[method].items())) CE = np.array(y)/y_cw + + # Add 95% confidence interval error bars (Clopper-Pearson exact method) ci = np.empty((2, len(y))) for i in range(len(y)): ci[:, i] = binomtest(y[i], y_cw[i]).proportion_ci() - plt.errorbar(x, CE*100, ci*100, fmt='-', label=method) + yerr = ci - CE + yerr[0] = -yerr[0] + plt.errorbar(x, CE*100, yerr*100, fmt='-', label=method) table.append([method, *CE*100]) print(tabulate(table, ["Method", *x], tablefmt="pipe", 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zip(*sorted(condorcet_paradox_counts.items())) CP = np.asarray(y) / n_elections # Likelihood of paradox -plt.plot(x, CP*100, '-', label='Simulation') + +# Add 95% confidence interval error bars (Clopper-Pearson exact method) +ci = np.empty((2, len(y))) +for i in range(len(y)): + ci[:, i] = binomtest(y[i], n_elections).proportion_ci() +yerr = ci - CP +yerr[0] = -yerr[0] +plt.errorbar(x, CP*100, yerr*100, fmt='-', label='Simulation') plt.legend() plt.grid(True, color='0.7', linestyle='-', which='major', axis='both')

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z7(j7f8&}bo2^Ro*p3narIc983MX#?dx}japzCRKs4n4k-q8*rzOoXJ%f1fzb=w1R< zNNRXjYfK!$;X6T46cMoT%5u1m{3jMm5^W}6V7VDVE3b{I&*;^s;GaI#P$yT*{K~%pG_Q*i From c93a525f3c3472927d186d21ee94e798acdba42f Mon Sep 17 00:00:00 2001 From: endolith Date: Wed, 20 Jul 2022 01:01:45 -0400 Subject: [PATCH 4/4] Add error bars to CE and CW likelihood examples These are also Bernoulli yes/no trials that produce binomial distributions. Update results images --- examples/merrill_1984_table_1_fig_1.py | 10 +++++++++- ..._Figure_1._25_voters,_10000_iterations.png | Bin 72909 -> 77856 bytes ...Wikipedia_Condorcet_paradox_likelihood.png | Bin 22511 -> 20205 bytes .../wikipedia_condorcet_paradox_likelihood.py | 10 +++++++++- 4 files changed, 18 insertions(+), 2 deletions(-) diff --git a/examples/merrill_1984_table_1_fig_1.py b/examples/merrill_1984_table_1_fig_1.py index db983bb..8861ff1 100644 --- a/examples/merrill_1984_table_1_fig_1.py +++ b/examples/merrill_1984_table_1_fig_1.py @@ -27,6 +27,7 @@ import time from collections import Counter import numpy as np +from scipy.stats import binomtest import matplotlib.pyplot as plt from tabulate import tabulate from elsim.methods import (fptp, runoff, irv, approval, borda, coombs, @@ -113,7 +114,14 @@ 'Black'): x, y = zip(*sorted(condorcet_winner_count[method].items())) CE = np.array(y)/y_cw - plt.plot(x, CE*100, '-', label=method) + + # Add 95% confidence interval error bars (Clopper-Pearson exact method) + ci = np.empty((2, len(y))) + for i in range(len(y)): + ci[:, i] = binomtest(y[i], y_cw[i]).proportion_ci() + yerr = ci - CE + yerr[0] = -yerr[0] + plt.errorbar(x, CE*100, yerr*100, fmt='-', label=method) table.append([method, *np.array(y)/y_cw*100]) # Likelihood that social utility maximizer is Condorcet Winner diff --git a/examples/results/Merrill_1984_Figure_1._25_voters,_10000_iterations.png b/examples/results/Merrill_1984_Figure_1._25_voters,_10000_iterations.png index f0f2aa6903a801bb2cb59f6ea3c77caaf48a6ae0..6f3e7b6a2e7e1788785617358ac3ff195f375ef6 100644 GIT binary patch literal 77856 zcmeEuWn5HW6zw1ql1d2D($XQFf`D{NmxOe8s|ZLdNOwzjH%OOA=g{3XH1CZ67a!io z_x1V1A2D<9J@@Q=)?RDvb0=6yK?)O%2n_;(U`oGxs{(;M`UHU>ETJNUzi^GtuYqrT z&Jvo=s&*fp-99*)LgYU<+gsZ?TU!`WxSBdTS=iZfvT(7mGf|j3JKH<)v$ER!?>Dg6 zIhwIfe>iCdCwXH3PRj`b!TtdMkMK(*&jNw~fk?l7qvoExx8UZCH!;_Cd}ysWbGC^7 z@(J}Bk&IXz?Kk>q^UUdWSiv;Xo>6DuGy*;zx}In@Y!Qdz<;y4A%~L}MO@2&Ts9fe- 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